If the secondary winding of the ideal transformer shown in the circuit of the figure has 40 turns, the number of turns in the primary winding for maximum power transfer to the 2 Ω resistor will be

Consider the following interconnection of the three LTI systems (Fig.1). $${h_1}(t)$$ , $${h_2}(t)$$ and $${h_3}(t)$$ are the impulse responses of these three LTI systems with $${H_1}(\omega )$$, $${H_2}(\omega )$$, and $${H_3}(\omega )$$ as their respective Fourier transforms. Given that $${h_1}\,(t)\, = \,{d \over {dt}}\left[ {{{\sin ({\omega _0}t)} \over {2\,\pi \,t}}} \right],{H_2}(\omega ) = \exp \left( {{{ - j2\pi \omega } \over {{\omega _0}}}} \right)$$
$${h_3}\,(t)\, = u(t)\,and\,x(t)\, = \,\sin \,2\,{\omega _0}t\, + \,\cos \,({\omega _0}t/2),$$ find the output y(t).

The Laplace transform of the periodioc function f(t) describe4d by the curve below, i.e., $$f\left( t \right) = \left\{ {\matrix{ {\sin \,t\,\,\,if\,\left( {2n - 1} \right)\pi \le t \le 2n\pi } \cr {0\,\,\,\,\,\,\,\,otherwise} \cr } } \right.$$
is _________. (fill in the blank), n is an integer.

If $$F\left( s \right) = L\left[ {f\left( t \right)} \right] = {K \over {\left( {s + 1} \right)\,\left( {{s^2} + 4} \right)}}$$ then $$\matrix{ {Lim\,f\,\left( t \right)} \cr {t \to \infty } \cr } $$ is given by